Finite Group Databasebeta

← #10591709

Subgroups — #10591709

Order 576.

542 conjugacy classes of subgroups; only the 108 normal subgroups are shown.

normal subgroups only

This group has 542 conjugacy classes of subgroups — too many to draw. Only its 108 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×#67C211yesmaximal, characteristic
2C2×#3757C211yesmaximal, characteristic, complemented
3C2×C3×#217C211yesmaximal, characteristic, complemented
4C2×Dic3⋊Dic3C211yesmaximal, characteristic, complemented
5C2×(C2×D3)⋊Dic3C211yesmaximal, characteristic, complemented
6C2×(C2×D3)⋊Dic3C211yesmaximal, characteristic, complemented
7#63579C211yesmaximal, characteristic, complemented
8#5222—31nomaximal
9#4961—31nomaximal
10C24×C32C2211yescharacteristic
11C23×C3×D3C2211yescharacteristic
12C22×C3⋊Dic3C2211yescharacteristic
13C22×C3⋊Dic3C2211yescharacteristic
14C2×C3×#67—22no—
15C2×C3×#67—22no—
16C2×C3×#67—22no—
17C22×C3×Dic3C2211yescharacteristic
18C22×C3×Dic3C2211yescharacteristic
19C22×C3×Dic3C2211yescharacteristic
20#3757—22no—
21C3×#217—22no—
22Dic3⋊Dic3—22no—
23(C2×D3)⋊Dic3—22no—
24(C2×D3)⋊Dic3—22no—
25C22×C3×D4—31no—
26C22×#67D311yescharacteristic
27C2×C3×C22⋊C4—31no—
28C2×#217—31no—
29C2×#217—31no—
30C2×#217—31no—
31C2×#217—31no—
32C2×#217—61no—
33C2×#217D311yescharacteristic, complemented
34C2×(C2×D3)⋊C4—31no—
35C2×(C2×D3)⋊C4—31no—
36C2×C4⋊Dic3—31no—
37C2×Dic3⋊C4—31no—
38C3⋊#95—31no—
39C3⋊#95—31no—
40C23×C32—21no—
41C23×C32—21no—
42C23×C32—21no—
43C23×C32—21no—
44C23×C32—21no—
45C23×C32—21no—
46C23×C32D412yes—
47C23×C32C2311yescharacteristic
48C22×C3×D3—21no—
49C22×C3×D3—21no—
50C22×C3×D3—21no—
51C22×C3×D3—21no—
52C22×C3×D3—21no—
53C22×C3×D3—21no—
54C22×C3×D3D412yes—
55C2×C3⋊Dic3—21no—
56C2×C3⋊Dic3—21no—
57C2×C3⋊Dic3—21no—
58C2×C3⋊Dic3—21no—
59C2×C3⋊Dic3—21no—
60C2×C3⋊Dic3—21no—
61C3×#67—28no—
62C2×C3×Dic3—21no—
63C2×C3×Dic3—21no—
64C2×C3×Dic3—21no—
65C2×C3×Dic3—21no—
66C2×C3×Dic3—21no—
67C2×C3×Dic3—21no—
68C2×C3×Dic3—21no—
69C2×C3×Dic3D412yes—
70C2×C3×Dic3D412yescomplemented
71#394—91noSylow
72C24×C3—21no—
73C24×C3—31no—
74C24×C3C2×D311yescharacteristic
75C24×C3C2×D311yescharacteristic
76C23×D3C2×D311yescharacteristic
77C2×C3×D4—62no—
78C2×C3×D4—62no—
79C2×C3×D4—62no—
80C22×C3×C4—31no—
81C22×C3×C4—31no—
82C22×C3×C4—31no—
83C2×#67—22no—
84C2×#67—22no—
85C2×#67—22no—
86C22×Dic3—31no—
87C22×Dic3—31no—
88C22×Dic3—31no—
89C22×Dic3—31no—
90C22×Dic3—61no—
91C22×Dic3—61no—
92C22×Dic3C2×D311yescharacteristic
93C22×Dic3C2×D311yescharacteristic
94C22×Dic3C2×D311yescharacteristic
95C3×C22⋊C4—62no—
96#217—22no—
97#217—62no—
98#217—64no—
99#217—62no—
100#217—62no—
101#217—62no—
102(C2×D3)⋊C4—62no—
103(C2×D3)⋊C4—62no—
104C4⋊Dic3—62no—
105Dic3⋊C4—62no—
106C22×C32—22no—
107C22×C32—22no—
108C22×C32—22no—
109C22×C32—41no—
110C22×C32—41no—
111C22×C32—41no—
112C22×C32—41no—
113C22×C32C2×D411yescharacteristic
114C22×C32C2×D411yescharacteristic
115C22×C32C2×D411yescharacteristic
116C22×C32C2×D411yescharacteristic
117C22×C32#4111yescharacteristic
118C22×C32#4111yescharacteristic
119C22×C32#4111yescharacteristic
120C2×C3×D3—22no—
121C2×C3×D3—22no—
122C2×C3×D3—22no—
123C2×C3×D3—41no—
124C2×C3×D3—41no—
125C2×C3×D3—41no—
126C2×C3×D3—41no—
127C3⋊Dic3—41no—
128C3⋊Dic3—41no—
129C3×Dic3—22no—
130C3×Dic3—22no—
131C3×Dic3—41no—
132C22×D4—31no—
133C2×C4⋊C4—91no—
134C2×C22⋊C4—31no—
135C2×C22⋊C4—91no—
136C2×C22⋊C4—91no—
137C2×C22⋊C4—91no—
138#95—91no—
139C23×C3—22no—
140C23×C3—22no—
141C23×C3—21no—
142C23×C3—21no—
143C23×C3—21no—
144C23×C3—21no—
145C23×C3—21no—
146C23×C3—21no—
147C23×C3—21no—
148C23×C3—21no—
149C23×C3—21no—
150C23×C3—21no—
151C23×C3—21no—
152C23×C3—21no—
153C23×C3—22no—
154C23×C3—21no—
155C23×C3—32no—
156C23×C3—41no—
157C23×C3—41no—
158C23×C3—41no—
159C23×C3—41no—
160C23×C3—61no—
161C23×C3—61no—
162C23×C3—61no—
163C23×C3—61no—
164C23×C3—61no—
165C23×C3—61no—
166C23×C3D1212yes—
167C23×C3#6712yes—
168C23×C3C22×D311yescharacteristic
169C23×C3C22×D311yescharacteristic
170C22×D3—21no—
171C22×D3—21no—
172C22×D3—21no—
173C22×D3—21no—
174C22×D3—21no—
175C22×D3—21no—
176C22×D3#6712yes—
177C3×D4—68no—
178C2×C3×C4—32no—
179C2×C3×C4—32no—
180C2×C3×C4—61no—
181C2×C3×C4—61no—
182C2×C3×C4—61no—
183C2×C3×C4—61no—
184C2×C3×C4—61no—
185C2×C3×C4—61no—
186C2×C3×C4—61no—
187#67—28no—
188C2×Dic3—21no—
189C2×Dic3—21no—
190C2×Dic3—21no—
191C2×Dic3—21no—
192C2×Dic3—21no—
193C2×Dic3—21no—
194C2×Dic3—21no—
195C2×Dic3—62no—
196C2×Dic3—62no—
197C2×Dic3—61no—
198C2×Dic3—61no—
199C2×Dic3—61no—
200C2×Dic3—61no—
201C2×Dic3—61no—
202C2×Dic3—61no—
203C2×Dic3—61no—
204C2×Dic3—61no—
205C2×Dic3—61no—
206C2×Dic3—61no—
207C2×Dic3—61no—
208C2×Dic3—61no—
209C2×Dic3—121no—
210C2×Dic3—121no—
211C2×Dic3—121no—
212C2×Dic3—121no—
213C2×Dic3#6712yes—
214C2×Dic3#6712yescomplemented
215C2×C32—42no—
216C2×C32#12011yescharacteristic
217C2×C32#12111yescharacteristic
218C2×C32#12111yescharacteristic
219C2×C32#12111yescharacteristic
220C2×C32#12311yescharacteristic
221C2×C32#12311yescharacteristic
222C2×C32#12411yescharacteristic
223C3×D3—42no—
224C24—31no—
225C24D3211yescharacteristic
226C2×D4—62no—
227C2×D4—62no—
228C2×D4—62no—
229C22×C4—31no—
230C22×C4—31no—
231C22×C4—31no—
232C22×C4—91no—
233C22×C4—91no—
234C4⋊C4—182no—
235C22⋊C4—62no—
236C22⋊C4—182no—
237C22⋊C4—182no—
238C22⋊C4—182no—
239C22×C3—24no—
240C22×C3—22no—
241C22×C3—22no—
242C22×C3—22no—
243C22×C3—22no—
244C22×C3—22no—
245C22×C3—22no—
246C22×C3—21no—
247C22×C3—21no—
248C22×C3—21no—
249C22×C3—21no—
250C22×C3—21no—
251C22×C3—21no—
252C22×C3—21no—
253C22×C3—42no—
254C22×C3—42no—
255C22×C3—42no—
256C22×C3—42no—
257C22×C3—42no—
258C22×C3—42no—
259C22×C3—41no—
260C22×C3—41no—
261C22×C3—41no—
262C22×C3—41no—
263C22×C3—41no—
264C22×C3—41no—
265C22×C3—41no—
266C22×C3—41no—
267C22×C3—62no—
268C22×C3—62no—
269C22×C3—62no—
270C22×C3—121no—
271C22×C3—121no—
272C22×C3—121no—
273C22×C3—121no—
274C22×C3C2×D1211yescharacteristic
275C22×C3#23511yescharacteristic
276C22×C3D3×D411yescharacteristic
277C22×C3D3×D411yescharacteristic
278C22×C3D3×D411yescharacteristic
279C22×C3#23711yescharacteristic
280C22×C3#23711yescharacteristic
281C22×C3#23711yescharacteristic
282C22×C3#23711yescharacteristic
283C22×C3#23711yescharacteristic
284C22×C3C2×#6711yescharacteristic
285C22×C3C2×#6711yescharacteristic
286C22×C3C2×#6711yescharacteristic
287C22×C3C2×#6711yescharacteristic
288C2×D3—22no—
289C2×D3—22no—
290C2×D3—22no—
291C2×D3—41no—
292C2×D3—41no—
293C2×D3—41no—
294C2×D3—41no—
295C3×C4—62no—
296C3×C4—62no—
297C3×C4—121no—
298Dic3—22no—
299Dic3—22no—
300Dic3—41no—
301Dic3—122no—
302Dic3—122no—
303Dic3—121no—
304Dic3—121no—
305Dic3—121no—
306Dic3—121no—
307C32#39411yescharacteristic, complemented, Sylow
308C23—21no—
309C23—21no—
310C23—21no—
311C23—21no—
312C23—21no—
313C23—21no—
314C23—32no—
315C23—61no—
316C23—61no—
317C23—61no—
318C23—61no—
319C23—61no—
320C23—61no—
321C23#62612yes—
322C23C2×D3211yescharacteristic
323D4—68no—
324C2×C4—32no—
325C2×C4—32no—
326C2×C4—61no—
327C2×C4—61no—
328C2×C4—61no—
329C2×C4—61no—
330C2×C4—61no—
331C2×C4—61no—
332C2×C4—61no—
333C2×C4—181no—
334C2×C4—181no—
335C2×C4—181no—
336C2×C4—181no—
337C2×C4—181no—
338C2×C4—181no—
339C2×C3—21no—
340C2×C3—21no—
341C2×C3—21no—
342C2×C3—21no—
343C2×C3—21no—
344C2×C3—21no—
345C2×C3—21no—
346C2×C3—44no—
347C2×C3—42no—
348C2×C3—42no—
349C2×C3—122no—
350C2×C3#88211yescharacteristic
351C2×C3#88311yescharacteristic
352C2×C3#88511yescharacteristic
353C2×C3#88611yescharacteristic
354C2×C3#93011yescharacteristic
355C2×C3#93511yescharacteristic
356C2×C3#93511yescharacteristic
357C2×C3#93611yescharacteristic
358C2×C3#93811yescharacteristic
359C2×C3#93811yescharacteristic
360C2×C3#93911yescharacteristic
361C2×C3#93911yescharacteristic
362C2×C3#93911yescharacteristic
363C2×C3#95311yescharacteristic
364D3—42no—
365C22—22no—
366C22—22no—
367C22—22no—
368C22—41no—
369C22—41no—
370C22—41no—
371C22—41no—
372C22—62no—
373C22—62no—
374C22—62no—
375C22—121no—
376C22—121no—
377C22—121no—
378C22—121no—
379C22#380411yescharacteristic
380C22#380511yescharacteristic
381C22#380511yescharacteristic
382C22C2×#62611yescharacteristic
383C22D3×#6711yescharacteristic
384C22D3×#6711yescharacteristic
385C22D3×#6711yescharacteristic
386C4—62no—
387C4—62no—
388C4—121no—
389C4—361no—
390C4—361no—
391C3—21no—
392C3#496111yescharacteristic, complemented
393C3#522211yescharacteristic, complemented
394C2—42no—
395C2—122no—
396C2#6395911yescharacteristic
397C2#6396211yescharacteristic
398C2#6396911yescharacteristic
399C2#6397011yescharacteristic
400C2#6397011yescharacteristic
401C2#6397711yescharacteristic
402C2#6397811yescharacteristic
subgroupG/H
center, FrattiniC23C2×D32
commutator, socle, cosocleC23×C32C23
FittingC24×C32C22
prime radicalC22×C3×#67C2
hypercenterC24D32
hypocenterC32#394
LCSG ▷ C23×C32 ▷ C32
UCSC1 ◁ C23 ◁ C24

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. C2×#3757C2semidirect
3. C2×C3×#217C2semidirect
4. C2×Dic3⋊Dic3C2semidirect
5. C2×(C2×D3)⋊Dic3C2semidirect
6. C2×(C2×D3)⋊Dic3C2semidirect
7. #63579C2semidirect
33. C2×#217D3semidirect
70. C2×C3×Dic3D4semidirect
214. C2×Dic3#67semidirect
307. C32#394semidirect
392. C3#4961semidirect
393. C3#5222semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C22×C3×#672
C2×#37572
C2×C3×#2172
C2×Dic3⋊Dic32
C2×(C2×D3)⋊Dic32
#635792
#52223
#49613